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User blog:KthulhuHimself/Comparing the norminals function to other already existing ones.
In this blog post, I'm going to be comparing and explaining the norminals function to other already existing function on the wiki (as the title suggests); which will include the first few numbers I will be coining which are defined using the norminals function. Due to the immense growth-rate of these functions, there won't be too much to compare it too; so feel free to ask me to add some additional functions for comparison below (i.e. specify which). Rayo(n) The first function, N<0>(n); is defined identically to FOST(10n), making it just a tad faster-growing than the Rayo function. Of course, the reason that it's defined just the same way as FOST(10n) and not FOST(n), is in order to have interesting results fir lower n, such as 4, 10, etc. Because the N<0>(n) function is defined identically to the FOST function, there is no reason to coin any specific numbers for it. I do not think that much more can be compared, so I'll move on. F7(n) Another notable function that compares to the norminals function is the one devised by none other than the Japanese googologist Fish. Now; because the Rayo hierarchy (Ra(n)) is defined in a practically identical way to N(n), it's easy to see that the growth-rates of the two coincide (to a rather close degree). Because F7(n) has a growth-rate approximately equal to that of Rgamma_0(n), and because Rgamma_0(n) has an approximately close growth rate to that of N(n); it's simple to see that F7(n) and N(n) are similar both in function and in growth-rate. As with above, because the N(n) function is very close to the F7(n) function, and there already is a number coined for the F7(n) function (Fish number 7), there is no reason to further coin any specific numbers for it. FOOT(n) And, of course; we reach the FOOT function. Commonly regarded as the strongest function on the wiki which isn't a naive extension; comparing it to the norminals function can most definitely be an interesting task. Now, on one of his earlier posts; Wojowu (LittlePeng9) had mentioned that the FOOT function was effectively the formalisation of what the RayoO'(n) function was intended to achieve; and because the actual FOOT function is defined quite differently, and in a slightly more difficult-to-compare way; I will be comparing the norminals function to the simplified version of the FOOT function. OOST(n) First, I will compare the norminals function to OOST, the first lagnuage/function relevant to the RayoO'(n) function. Because OOST is effectively defined as Rayoa with no limit on a, or in other words, put into context; O is defined as the "successor" to all ordinals (in the same sense that <0,1> is); from this, it is simple to see why N<0,1>(n) is pretty much defined the same way as OOST, making it a good comparison. Number! Of course, it's about time I've coined a number here, so just for fun; let's define - K1 = N<0,1>10(100) Well; it's big... but we can do bigger. Regarding other instances of O Later in his post, Wojowu goes on to define O+1, O+O, O^2, etc. in a rather identical way to how I define <1,1>, <<0,1>,1>, <<<...<0,1>...,1>,1>,1> (with <0,1> nestings), etc.; showing that O is effectively the same as <0,1> in pretty much all senses. At a certain point; Wojowu goes on to define O1, as the collection of all reordering of O; just the same way <0,2> is the successor of all reordering of ; O2 as being the collection of all reordering of O1 (identically to <0,3>), OO as being the collection of all reordering of Oa (identically to <0,<0,1>>) etc. O'OST and FOOT Of course, Wojowu finally defines O' as being the limit of anything using Oa, just the same way <0,0,1> is defined in regards to <0,a>. Needless to say, this means that N<0,0,1>(n) is the best comparison to FOOT(n); as the two are defined almost identically. Another number! Let's define: K2 = ]N[1(100) Needless to say, it's big; very big (and yes, it's larger than BIG FOOT). Onwards? Of course, the norminals function extends beyond this basal form, creating functions both more powerful and more interesting than its forms discussed here. A simple example of this would be N<0,0,0,1>(n), a function alrady ''far ''more powerful than N<0,0,1>(n) (and hence FOOT(n)); let alone the even stronger functions given rise to on my "Norminals revisited" post. Category:Blog posts